Integrated circuits are formed of many layers of different materials, which layers are patterned so as to form desired structures that interact with one another according to predetermined designs. Thus, it is of vital importance that many of these layers be formed to very exacting tolerances, such as in their shape, thickness, and composition. If the various structures so formed during the integrated circuit fabrication process are not precisely formed, then the integrated circuit tends to not function in the intended manner, and may not function at all.
As the term is used herein, “integrated circuit” includes devices such as those formed on monolithic semiconducting substrates, such as those formed of group IV materials like silicon or germanium, or group III-V compounds like gallium arsenide, or mixtures of such materials. The term includes all types of devices formed, such as memory and logic, and all designs of such devices, such as MOS and bipolar. The term also comprehends applications such as flat panel displays, solar cells, and charge coupled devices.
Because the layers of which integrated circuits are formed are so thin and patterned to be so small, they cannot be inspected without the aid of instrumentation. The precision of the instrumentation used is, therefore, vital to the successful production of integrated circuits. Thus, any improvement that can be made in the accuracy of such instrumentation is a boon to the integrated circuit fabrication industry. In addition, any improvement in the speed at which such instrumentation can take its readings is also of benefit to the industry, as such speed enhancements tend to reduce the production bottlenecks at inspection steps, or alternately allow for the inspection of a greater number of integrated circuits at such inspection steps.
Spectral ellipsometers and dual beam spectrophotometers are typically used to measure properties such as thickness and refractive index of individual layers within a multilayered film stack. Such instruments work by directing one or more beams of light toward the surface of the film stack, and then sensing the properties of the light as it is variously reflected off of the different surfaces of the individual layers within the film stack. By adjusting the properties of the incident beam of light, and detecting the associated changes in the reflected beam of light, the properties of the film stack, such as the materials of which the various layers are formed and the thicknesses to which they are formed, can be determined. Such methods typically involve solving Maxwell's equations, which provide a model for such systems.
This film measurement process can be broken down into two basic steps, being 1) the measurement of the properties of the reflected light beam, and 2) the mathematical fitting of reflectance property values from Maxwell's equations, which are solved or estimated, to the measured results attained in step 1. Step 2 typically consists of the iterated steps of computing one or more theoretical values by plugging estimates of the film stack parameters, such as thickness and refractive index, into the model film stack equations, comparing the theoretical values obtained to the actual measured property values of the reflected beam of light, and if the theoretical values and the measured values do not agree to within a desired tolerance, then adjusting the estimated film stack parameters and recomputing the theoretical values.
This process is performed again and again, each time making some adjustment to the estimated film stack parameters that are fed into the model, until the theoretical values computed by the model agree with the actual measured values within the desired precision limits. When this agreement is attained, then there is some confidence that the estimated film stack parameters that were used to produce the theoretical values are very nearly the same as the actual film stack parameters.
For film stacks containing non-isotropic, inhomogeneous layers, such as layers containing metal patterns or gratings, the theoretical values are commonly generated by mathematical models based on either the rigorous coupled wave analysis or modal method.
The reflectance equations that are derived for film stacks that generate multiple plane waves traveling at different angles are typically formulated using at least one of the S matrix algorithm or the R matrix algorithm, as understood by those with skill in the art. However, these algorithms are computationally burdensome, and require a longer than desirable period of time—and greater than desirable computational resources—to solve. Another well-known alternate formulation for computing the reflectance is the T matrix algorithm. Although this algorithm is more computationally efficient, it is highly susceptible to computational overflow error and is therefore seldom used in practical applications.
What is needed is a method to improve the efficiency of the electromagnetic field computation of multilayer structures containing patterned dielectric structures.